24. Cosmological parameters 1 24. Cosmological Parameters Updated October 2017, by O. Lahav (University College London) and A.R. Liddle (University of Edinburgh). 24.1. Parametrizing the Universe Rapid advances in observational cosmology have led to the establishment of a precision cosmological model, with many of the key cosmological parameters determined to one or two significant figure accuracy. Particularly prominent are measurements of cosmic microwave background (CMB) anisotropies, with the highest precision observations being those of the Planck Satellite [1,2] which supersede the iconic WMAP results [3,4]. However the most accurate model of the Universe requires consideration of a range of observations, with complementary probes providing consistency checks, lifting parameter degeneracies, and enabling the strongest constraints to be placed. The term ‘cosmological parameters’ is forever increasing in its scope, and nowadays often includes the parameterization of some functions, as well as simple numbers describing properties of the Universe. The original usage referred to the parameters describing the global dynamics of the Universe, such as its expansion rate and curvature. Also now of great interest is how the matter budget of the Universe is built up from its constituents: baryons, photons, neutrinos, dark matter, and dark energy. We need to describe the nature of perturbations in the Universe, through global statistical descriptors such as the matter and radiation power spectra. There may also be parameters describing the physical state of the Universe, such as the ionization fraction as a function of time during the era since recombination. Typical comparisons of cosmological models with observational data now feature between five and ten parameters. 24.1.1. The global description of the Universe : Ordinarily, the Universe is taken to be a perturbed Robertson–Walker space-time with dynamics governed by Einstein’s equations. This is described in detail in the Big-Bang Cosmology chapter in this volume. Using the density parameters Ωi for the various matter species and ΩΛ for the cosmological constant, the Friedmann equation can be written k Ω + Ω − 1 = , (24.1) i Λ R2H2 Xi where the sum is over all the different species of material in the Universe. This equation applies at any epoch, but later in this article we will use the symbols Ωi and ΩΛ to refer to the present-epoch values. The complete present-epoch state of the homogeneous Universe can be described by giving the current-epoch values of all the density parameters and the Hubble constant h −1 −1 (the present-day Hubble parameter being written H0 = 100h km s Mpc ). A typical collection would be baryons Ωb, photons Ωγ, neutrinos Ων , and cold dark matter Ωc (given charge neutrality, the electron density is guaranteed to be too small to be worth considering separately and is effectively included with the baryons). The spatial curvature can then be determined from the other parameters using Eq. (24.1). The total present matter density Ωm = Ωc + Ωb may be used in place of the cold dark matter density Ωc. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) June 5, 2018 19:56 2 24. Cosmological parameters These parameters also allow us to track the history of the Universe, at least back until an epoch where interactions allow interchanges between the densities of the different species; this is believed to have last happened at neutrino decoupling, shortly before Big Bang Nucleosynthesis (BBN). To probe further back into the Universe’s history requires assumptions about particle interactions, and perhaps about the nature of physical laws themselves. The standard neutrino sector has three flavors. For neutrinos of mass in the range 5 × 10−4 eV to 1 MeV, the density parameter in neutrinos is predicted to be 2 mν Ων h = , (24.2) 93P.14eV where the sum is over all families with mass in that range (higher masses need a more sophisticated calculation). We use units with c = 1 throughout. Results on atmospheric and Solar neutrino oscillations [5] imply non-zero mass-squared differences between the three neutrino flavors. These oscillation experiments cannot tell us the absolute neutrino masses, but within the simple assumption of a mass hierarchy suggest a lower limit of approximately 0.06 eV for the sum of the neutrino masses (see the Neutrino chapter). Even a mass this small has a potentially observable effect on the formation of structure, as neutrino free-streaming damps the growth of perturbations. Analyses commonly now either assume a neutrino mass sum fixed at this lower limit, or allow the neutrino mass sum as a variable parameter. To date there is no decisive evidence of any effects from either neutrino masses or an otherwise non-standard neutrino sector, and observations impose quite stringent limits; see the Neutrinos in Cosmology section. However, we note that the inclusion of the neutrino mass sum as a free parameter can affect the derived values of other cosmological parameters. 24.1.2. Inflation and perturbations : A complete model of the Universe should include a description of deviations from homogeneity, at least in a statistical way. Indeed, some of the most powerful probes of the parameters described above come from the evolution of perturbations, so their study is naturally intertwined with the determination of cosmological parameters. There are many different notations used to describe the perturbations, both in terms of the quantity used to describe the perturbations and the definition of the statistical measure. We use the dimensionless power spectrum ∆2 as defined in the Big Bang Cosmology section (also denoted P in some of the literature). If the perturbations obey Gaussian statistics, the power spectrum provides a complete description of their properties. From a theoretical perspective, a useful quantity to describe the perturbations is the curvature perturbation R, which measures the spatial curvature of a comoving slicing of the space-time. A simple case is the Harrison–Zeldovich spectrum, which corresponds 2 to a constant ∆R. More generally, one can approximate the spectrum by a power-law, writing ns−1 2 2 k ∆R(k) = ∆R(k∗) , (24.3) ·k∗ ¸ June 5, 2018 19:56 24. Cosmological parameters 3 where ns is known as the spectral index, always defined so that ns = 1 for the Harrison–Zeldovich spectrum, and k∗ is an arbitrarily chosen scale. The initial spectrum, defined at some early epoch of the Universe’s history, is usually taken to have a simple form such as this power law, and we will see that observations require ns close to one. Subsequent evolution will modify the spectrum from its initial form. The simplest mechanism for generating the observed perturbations is the inflationary cosmology, which posits a period of accelerated expansion in the Universe’s early stages [6,7]. It is a useful working hypothesis that this is the sole mechanism for generating perturbations, and it may further be assumed to be the simplest class of inflationary model, where the dynamics are equivalent to that of a single scalar field φ with canonical kinetic energy slowly rolling on a potential V (φ). One may seek to verify that this simple picture can match observations and to determine the properties of V (φ) from the observational data. Alternatively, more complicated models, perhaps motivated by contemporary fundamental physics ideas, may be tested on a model-by-model basis (see more in the Inflation chapter in this volume). Inflation generates perturbations through the amplification of quantum fluctuations, which are stretched to astrophysical scales by the rapid expansion. The simplest models generate two types, density perturbations that come from fluctuations in the scalar field and its corresponding scalar metric perturbation, and gravitational waves that are tensor metric fluctuations. The former experience gravitational instability and lead to structure formation, while the latter can influence the CMB anisotropies. Defining slow-roll parameters, with primes indicating derivatives with respect to the scalar field, as m2 V ′ 2 m2 V ′′ ǫ = Pl , η = Pl , (24.4) 16π µ V ¶ 8π V which should satisfy ǫ, |η| ≪ 1, the spectra can be computed using the slow-roll approximation as 2 8 V 2 128 ∆R(k) ≃ 4 ¯ , ∆t (k) ≃ 4 V ¯ . (24.5) 3m ǫ ¯ 3m ¯ Pl ¯k=aH Pl ¯k=aH ¯ ¯ ¯ ¯ In each case, the expressions on the right-hand side are to be evaluated when the scale k is equal to the Hubble radius during inflation. The symbol ‘≃’ here indicates use of the slow-roll approximation, which is expected to be accurate to a few percent or better. From these expressions, we can compute the spectral indices [8]: ns ≃ 1 − 6ǫ +2η ; nt ≃ −2ǫ. (24.6) Another useful quantity is the ratio of the two spectra, defined by 2 ∆t (k∗) r ≡ 2 . (24.7) ∆R(k∗) June 5, 2018 19:56 4 24. Cosmological parameters We have r ≃ 16ǫ ≃ −8nt , (24.8) which is known as the consistency equation. One could consider corrections to the power-law approximation, which we discuss later. However, for now we make the working assumption that the spectra can be approximated by such power laws. The consistency equation shows that r and nt are not independent parameters, and so the simplest inflation models give initial conditions described by three 2 parameters, usually taken as ∆R, ns, and r, all to be evaluated at some scale k∗, usually the ‘statistical center’ of the range explored by the data. Alternatively, one could use the parametrization V , ǫ, and η, all evaluated at a point on the putative inflationary potential. After the perturbations are created in the early Universe, they undergo a complex evolution up until the time they are observed in the present Universe. When the perturbations are small, this can be accurately followed using a linear theory numerical code such as CAMB or CLASS [9].